Half step

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In western music, half step is one twelfth of an octave.

Formula for a Half Step

Since an octave has a frequency ratio of 2, a half-step has a frequency ratio of 2^(1/12), or approximately 1.0595.

For example, if the note A has a frequency of 440 Hz, then one half-step up (A# or Bb) is 440*1.0595 = 466.2 Hz. One half-step down (G# or Ab) is 440/1.0595 = 415.3 Hz.

(Just like an octave, a half-step interval is a ratio obtained through division, not a difference obtained through subtraction. One octave above 440 Hz is 880 Hz, while one octave below is 220 Hz. The size of the octaves in Hz is different, but the ratio is the same.)

Correctness of this formula

For music played on a modern piano, guitar, or electronic keyboard, this formula is exact, as these instruments use an equal-tempered scale.

However, notes sung by a voice or played on an instrument in which the instrumentalist can dynamically adjust the pitch of the note, may vary slightly from the exact tunings implied by an equal-tempered scale. In particular, two voices or two instruments playing in harmony will tend to drift towards ratios of small natural numbers: for example 3:2, 4:3, and 5:4.

In all human music, harmony is formed by ratios of small natural numbers: 550 Hz is harmonious with 440 Hz. This is universal. In Western music (99% of all music composed in the last 50 years), the octave is divided into 12 equal parts. This is completely arbitrary, there is nothing universal about the 12 parts. Conveniently, approximations to most of the ratios of small natural numbers can be found in that division into 12, so almost all universal harmonies are possible.

See the Wikipedia article Semitone for more information.

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